I agree that it could easily be a growth spurt for a boy at age 10. Get things checked out with someone other than his current doctor, but don't jump on the bandwagon of drugs. They often don't fix much and are just a bandaid. I am glad that you are hesitant.
Doctors often will overreact. When one of my kids was a baby-young child, doctors were worried about weight & had me coming in constantly. It was all about their comparison charts and what was considered "normal." They forget that there is also big and small. Not everyone is built the same. Metabolisms vary widely. It was ridiculous. What added to it was that we as parents were small too. To this day, my adult child is still on the small side, but is healthier than most young adults out there.
We have a niece who was even more thin. To this day, as an adult, she is still very thin/skinny. It is just her metabolism.
I was always told by wise older moms that if the child is relatively healthy, eating well, strong, happy and active (mine was SO active), that there is nothing to worry about. You don't sound overly worried, but I can see that all of the fuss is messing with you. You referred to having the same things happen when you were young, and this makes me suspect that he is normal for your family.
What I would say is to make sure that he is active, eating a real food, whole foods diet (no white & no processed foods, but stick to traditional foods, minimal additives), and keep him active.
What the doctor recommended in the form of high carbs will only lead to unhealthy fat gain, when what is much more desirable is muscle with a bit of healthy fat gain. Eating constantly will not give his digestive tract a chance to rest either, which is important. Consuming fat does not lead to fat either. Don't be afraid of adding healthy minimally processed fats (olive, grapeseed or coconut oils, some animal fats including real butter, avocado...canola is highly processed). If there are foods that physically bother him, cut them out and find a healthy alternative. This is a less alarmist way to approach everything.
On that note, and not to alarm you, but just an idea to consider that most people don't think of...worms can be something that infect people and can cause lack of weight gain. Old world cultures often regularly ate certain foods to help eliminate these. We don't do this in North America. It might be worth asking a natural practitioner about. Their ways will be much more gentle on the body than conventional. I do know that an infusion (strong tea made with cold water) of broadleaf plantain is good. Adding mint helps perk up the flavor. Plantain is very healthy regardless. Hedleys might carry it. It is past the plant's DIY harvest time now. Plantain is only one of many herbs that can help if this is an issue.
About the anxiety, all of the intense "concern" won't help. It sends the message to your son that he has something wrong with him, he needs to be fixed and so on. This in itself could be a large part of his experiencing anxiety. If the anxiety is something that you feel needs more attention, there are many safe herbs that can be used for anxiety with way less side effects than meds. Also figuring out what is bothering him in his environment (stresses from school, people, games, tv, processed foods, other sources) and trying to figure out how to lessen these stresses might help a lot.
I hope that you can both find peace on the other side of all this noise. I, for one, would be interested in what the conclusion might be. Do your own research, consider all avenues and weigh it all. Hang in there.